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20 CHAPTER 1 Security and Vulnerability of Cyber-Physical Infrastructure Networks us briefly describe the synchronization model and initial-condition estimation problem, before summarizing results of the performance-analysis [8]. Most broadly, we consider a linear time- invariant (LTI) dynamics specified by a weighted and directed graph G . Precisely, let us consider a graph = (V , E : ), where the vertex set V con- tains n elements labeled 1, . . . , n, the edge set E contains q directed edges or ordered pairs of dis- tinct vertices, and each edge (i , j ) in E has asso- ciated with it a positive weight ij as given in the weight set . To describe the diffusive net- work dynamics, we find it convenient to specify an m × m diffusion matrix L from the graph , as follows: the hope of providing the reader with a brief tuto- rial in the network estimation methods. Initial-Condition Estimation on a Laplacian Network System. Synchronization or diffu- sion phenomena are common in modern cyber- physical networks: for instance, in the cyber world, some agreement and flooding algorithms have this form, whereas numerous physical dynamics such as electric power and disease- spread dynamics can also be well approximated in this way [10, 23, 24]. Thus, we are moti- vated to study security in network synchroniza- tion processes, by characterizing the performance of structure or dynamics estimator that use noisy local data. That is, we are motivated to character- ize the ability of an adversary to estimate aspects of a synchronization dynamics, from local and noisy measurements. In fact, in our recent work, we have pursued a graph-theoretic analysis of state inference in discrete network synchroniza- We set L ij equal to - ij , for each ordered pair (i , j ) E . We set L ij , i = j , equal to 0 otherwise. We choose L ii = - n =1,j =i L ij . That is, we j